76 OPTICS LETTERS / Vol. 6, No. 2 / February 1981

Observation of amplified reflection through degenerate four-wave mixing at CO2 laser wavelengths in germanium

D. E. Watkins, C. R. Phipps, Jr., and S. J. Thomas

Los Alamos Scientific Laboratory, University of California, Los Alamos, New Mexico 87545

Received September 15, 1980

We report the first known observation of amplified reflection through degenerate four-wave mixing at 10.6 ,m.Reflectivities of over 100% are reported in both n-type and p-type germanium at pump intensities of about 100MW/cm 2. The maximum reflectivity reported is 800% in p-type Ge.

We describe the first known observation of amplifiedphase-conjugate reflection at 10.6 Atm through degen-erate four-wave mixing in an optically induced plasmain germanium. Phase-conjugate reflectivities of100-800% have been observed in several samples of Gepumped at intensities of about 100 MW/cm2.

Degenerate four-wave mixing (DFWM) in opticallyinduced plasmas has previously been reported. Jain etal.,1 and earlier Woerdman,2 reported resonantly en-hanced DFWM at 1.06 Atm, near the band gap of Si.Both Khan3 and Jain4 and their co-workers have re-ported similar effects in Hgl-,CdxTe at 10 l.m. In thismaterial, the band gap can be tuned to resonance. Inboth these materials, a large concentration of free car-riers is generated by single-photon absorption. Thisfree-carrier plasma exhibits the spatial variations of theinterfering probe and pump beams, perturbing theindex of refraction to create the gratings that scatter thepump beams and generate the phase-conjugate beam.In our case, the scattering mechanism is the same, but,since we are operating far from the band gap, formationof the optically induced plasma involves a multipho-ton-absorption process.

The plasma-formation process has been the subjectof intensive study by two of the authors,5 by Yuen etal., 6 and by Danileiko et al. 7 Hellwarth first suggestedto us that this process could be used in DFWM.8

Our experimental technique for measuring phase-conjugate reflectivity has been described earlier.9 Inthis work we employed an additional detector to observeamplification of the probe beam and used a 1.5-nsec,10.59-Am laser pulse. The maximum intensity avail-able in each pump beam was 150 MW/cm 2 , above thesurface-damage threshold for the antireflection-coatedGe in the counterpropagating-pump geometry. Theexternal angle between the probe and the pump beamswas about 150, corresponding to about 40 inside the Ge.The samples studied were between 3 and 20 mm long,either of optical grade or slightly p-type, and eithersingle-crystal or polycrystalline.

Figure 1 shows the results of phase-conjugation ex-periments in two samples of Ge, each 3 mm long. Onewas optical-grade Ge, and the other was p-type with asmall-signal intensity transmission of 0.43 corre-sponding to an acceptor concentration of 6 X 1015 cm- 3 .The plasma-formation process occurs at intensities

above about 80 MW/cm 2 in the counterpropagating-pump configuration, and in this regime the phase-conjugate reflectivity exhibited a dramatic increase.The reflectivity varied approximately as the 11th powerof intensity over the range 80 < I < 120 MW/cm2. Thepresence of the free-carrier plasma was confirmed byapplying small dc voltage across the sample and mea-suring the conductivity. We observed enhanced re-flectivity only in the presence of an increase in con-ductivity. Amplification of the forward probe beamwas also observed in the optical-grade sample, and wemeasured an increase of about 80% in this signal, cor-related to the large phase-conjugate reflectivities.Attenuation of the probe beam by the medium accountsfor any discrepancy.

We can estimate the reflectivity by calculating thechange in index of refraction that is due to plasma for-mation and by using the formula for reflectivity givenby Yariv and Pepper.'0 To do this, we must first obtainthe density, which is accomplished experimentally, andthen calculate the change in index of refraction, takinginto careful consideration the effects of each carrier typein both cases.

01 0

-1I10

> -24-'1 0

e -3,,10

-410

N

Pump Intensity (Mw/cm2 )

Fig. 1. Phase-conjugate reflectivity in 3-mm polycrystallinesamples of p-type (stars) and optical grade (crosses) Ge. Thelines are the models discussed in the text.

0146-9592/81/020076-03$0.50/0 © 1981, Optical Society of America

-510Lt EM

February 1981 / Vol. 6, No. 2 / OPTICS LETTERS 77

The free-carrier density is determined experimentallyby carefully time resolving the incident and transmittedsignals obtained above the plasma threshold in Ge anddeducing the free-carrier density by use of a detailedmodel that we developed to relate this quantity to theinstantaneous, intensity-dependent transmission.Although a more detailed description of this work willappear in a future publication, we present the relevantaspects of it here.

When volumetric photoionization occurs at CO2wavelengths, three types of carrier are produced, eachaffecting absorption in a different way, and three widelydisparate time scales are involved in the evolution oftheir densities. The longest time scale is that for re-combination, which is 103 to 106 times longer than thelaser-signal duration," ensuring that charge neutralitycontrols the densities of new holes and electrons created;ANh = ANe.

The next shorter scale is that for evolution of theplasma-controlled transmission, which is observed tobe of the order of the laser-pulse duration, that is, about300 psec. Much faster yet (about 2 psec) is the timerequired for partition of the local total density of holesbetween the heavy- and light-hole bands.12 The pri-mary effect of this redistribution is bleaching of theassociated interband transition that is responsible formost of the absorption at 10.6 gm. A secondary effectnot normally considered is an important change in thenet Drude-Zener absorption of the free-carrier en-semble, which is due mainly to changes in the light-holedensity, since these have extremely small effective mass.Given the total density of holes, N = ANh + NO, whichincludes the initial acceptor density No, the partitionbetween the total heavy- and light-hole densities is

NL (I) = (N/2) [1 - (1 + I/II)-1/2],NH(I) = (N/2)[1 + (1 + I/II)-1/2]. (1)

In Eq. (1), i, is the saturation intensity13 of the inter-band transition (about 3.2 MW/cm2), and the form ofthe expression is due to the fact that the transition isinhomogeneously broadened.12

Ignoring only the Drude-Zener absorption compo-nent that is due to the free heavy holes (it is extremelysmall), we write for the total local absorption coeffi-cient:

Atot = aLp + Neae + NLqL + (NH - NL) f1 2 , (2)

where ap, Ue and aL, and a12 are, respectively, the re-sidual absorption coefficient due to lattice vibrations,the Drude-Zener cross section for free electrons and forlight holes, and the interband cross section. Wetake14"15 ap = 0.035 cm-1, a12 = 6.8 A2, (rYL/a12) = 0.085,and ((e/f1 2 ) = 0.044. Ne = ANe + Ni 2/No is the totalfree-electron density, where Ni is the mass-actiondensity, 2.3 X 1013 cm- 3 in room-temperature Ge."1 Wealso obtain for the change in index of refraction

An 1 47re2 Njlmj*2n0, meW 2 F-

J

27re2 N{Ne/Nmc*

flameOw)2

+ [1 + (1 + I/is)- 1/2]/2mH*+ [1 - (1 + I/Is )-/ 2]/2mL*}, (3)

where no is the linear index of refraction; co is the laserfrequency; e is the charge; me is the mass of an electron;and mc*/me = 0.12, mH*/me = 0.31, and mL*/me =0.044 are the conductivity effective masses for theconduction, heavy-hole, and light-hole bands, respec-tively.'6

These results facilitate two procedures. In the first,a bivariate reverse iteration is performed, in whichRigrod theory with loss17 is applied to experimentaltime-resolved transmission data, using Eq. (2) to obtainthe appropriate fixed and bleachable components of theabsorption to determine what average excess carrierdensity would have provided the observed transmissionwith the appropriate spatial variation of the heavy- tolight-hole partition. When this is done, for example,with samples of intrinsic Ge of three different lengths,results such as those in Fig. 2 are obtained. In the fig-ure, the threshold for strong photoionization in theunidirectional beam is seen to be about 200 MW/cm-2.Above this intensity, excess carrier density varies withintensity to the 5.5 power.

In the second procedure, we use the density-versus-intensity profile to estimate the change in refractiveindex and thus the expected reflectivity. For N = 2.5X 1015 cm- 3 and I/IS = 125, we obtain An = -6.4 X10-4. From Yariv and Pepper,' 0 the reflectivity pro-duced by this refractive-index grating is given by R =tan 2(k0 AnL), where ho is the free-space wave numberand L is the interaction length. For L = 3 mm, we ob-tain R = 4.8, in reasonable agreement with the experi-ment. Although this calculation ignores attenuationby the medium, one should note that a factor-of-2 in-crease in the argument of the tangent would result ininfinite gain, so the fit is much more sensitive to theassumed average value of An than to the approximately50% attenuation experienced by the reflected signal.This model also predicts the intensity dependence ofthe reflectivity to be approximately Ill over the range80 < I < 120 MW/cm2 , consistent with the experimentalresults.

Below the plasma-formation threshold, the phase-conjugate reflectivity of the optical-grade Ge is modeledby R = tan 2 (koAnL), where An = (127rX3/n)EfEb andX3 = 3.2 X 10-' esu.18 For p-type Ge, below threshold,

1016

a;

o 2 *

fId4.

100 200 300 400 500PEAK INPUT INTENSITY (MW cm-

2)

Fig. 2. Peak free-carrier density versus peak input intensityin intrinsic Ge with three different sample lengths. Circles,5.7 cm; triangles, 9.7 cm; stars, 14.2 cm. The trend line rep-resents 15-5.

78 OPTICS LETTERS / Vol. 6, No. 2 / February 1981

10

4 I '

U -2-10

-31 0+

-41 0 +

Pump Intensity (mw/cm2)

Fig. 3. Phase-conjugate reflectivity in a 20-mm single-crystalsample of optical-grade Ge.

resonant effects that are due to saturable absorption bythe holes enhance the reflectivity relative to that foroptical grade, as shown in Fig. 1. The basis for our p-type model is similar to that of Abrams and Lind,19except that we have included the inhomogeneousbroadening of the transition and the attenuation of thepump beams by the medium. There are no free pa-rameters in this model, and it is valid when, as in thiscase, the contribution of the saturable absorber to thereflectivity is much greater than that of the host Ge.The details of this model will be presented later.20

Figure 3 presents data taken using a 2-cm sample ofsingle-crystal optical-grade Ge. The data are fittedwith Yariv and Pepper's model with X3 = 2.5 X 10-1"esu. Here, the free-carrier enhancement of the reflec-tivity is limited by absorption of the pump beams by theplasma to a maximum value of 400%.

We also observed the enhancement effect in a 6-mmsample of lightly doped p-type Ge (small-signal trans-mission of 0.70) and measured a maximum phase-con-jugate reflectivity of 800%.

In conclusion, we have presented the first knownobservation of amplified reflection at CO2 wavelengthsthrough degenerate four-wave mixing in an opticallyinduced plasma in Ge. The maximum reflectivity ob-

served was 800%. Observations were made in bothslightly n-type (optical-grade) Ge and p-type Ge.Pump powers near 100 MW/cm 2 were necessary to ob-serve the enhancement of the phase-conjugate reflec-tivity by the free-carrier formation process in Ge.

The authors gratefully acknowledge the support andencouragement of J. F. Figueira and B. J. Feldman andthe technical assistance of R. F. Harrison and R. Shaw.This research was performed under the auspices of theU.S. Department of Energy.

References

1. R. K. Jain and M. B. Klein, Appl. Phys. Lett. 35, 454(1979); R. K. Jain, M. B. Klein, and R. C. Lind, Opt. Lett.4, 328 (1979).

2. J. P. Woerdman, Opt. Commn. 2, 212 (1970).3. M. A. Khan, P. W. Kruse, and J. F. Ready, Opt. Lett. 5,

261 (1980).4. R. K. Jain and D. G. Steel, Appl. Phys. Lett. 37, 1

(1980).5. C. R. Phipps, Jr., and S. J. Thomas, unpublished.6. S. Y. Yuen et al., Opt. Commun. 28, 237 (1979); S. Y.

Yuen, R. L. Aggarwal, and B. Lax, J. Appl. Phys. 51,1146(1980).

7. Y. Danileiko et al., Reprint No.10 (P.N. Lebedev PhysicalInstitute, Moscow, 1977).

8. R. W. Hellwarth, Electronics Sciences Laboratory, Uni-versity of Southern California, Los Angeles, Calif. 90007,personal communication, 1978.

9. D. E. Watkins, J. F. Figueira, and S. J. Thomas, Opt. Lett.5, 169 (1980).

10. A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 (1977).11. R. A. Smith, Semiconductors (Cambridge U. Press,

Cambridge, 1968), pp. 350-355.12. F. Keilmann, IEEE J. Quantum Electron. QE-12, 592

(1976).13. C. R. Phipps, Jr., and S. J. Thomas, Opt. Lett. 1, 93

(1977).14. E. D. Capron and 0. L. Brill, Appl. Opt. 12, 569 (1973).15. H. B. Briggs and R. C. Fletcher, Phys. Rev. 91, 1342

(1953).16. B. Lax and J. G. Mavroides, Phys. Rev. 100, 1650

(1955).17. W. W. Rigrod, J. Appl. Phys. 34,3602 (1963).18. D. E. Watkins, C. R. Phipps, Jr., and S. J. Thomas, Opt.

Lett. 5, 248 (1980).19. R. L. Abrams and R. C. Lind, Opt. Lett. 2,94,3, 205(E)

(1978).20. W. W. Rigrod and D. E. Watkins, unpublished.